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JCATiONS  STATE  NORMAL  SCHOOL  CH1CO,  CALIFORNIA 

NO.  4  MARCH,  1909 


IJ  'N      PJLu 


ALVA  W,   STAMPER 


State  Normal  School,  Chico,  California 
Publications 


BULLETINS 

No   1.    School  Gardens  for  California  Schools.     B.  M.  DAVIS,  PH.  D. 

80  pages.  30c. 
No.  2.  A  Guide  to  the  Birds  of  the  Pacific  Coast.  C.  A.  STEBBINS. 

24  pages.  25c. 
No.  3.  The  Teaching  of  Arithmetic.  ALVA  W.  STAMPER,  PH.  D. 

80  pages.  50c. 
No.  4.  Lesson  Plans  in  Arithmetic.  ALVA  W.  STAMPER,  PH.  D. 

17  pages.  20c.   ( Also  contained  in  Bulletin  No.  3.  ) 

OTHER    PUBLICATIONS 

One  Hundred  Experiments  in  Elementary  Agriculture  for  California 
Schools.  RILEY  0.  JOHNSON,  A.  B.  4'2  \MW*.  :'.0c. 

Course  in  Nature  Study  and  Elementary  Agriculture  for  the  Un- 
graded Schools  of  California.  RILEY  0.  JOHNSON,  A.  B. 
8  pages.  lOc. 

Outline  of  Geography  Methods.    C.  K.  STUDLEY,  A.  B.  30  pages.  35c. 

Physical  Geography  Outline.     C.  K.  STUDLEY,  A.  B.     10  pages.   15c. 

Nature  Study  and  Agriculture  for  California  Schools.  C.  A.  STEBBINS. 
160  pages.  40c. 


LESSON  PLANS  IN  ARITHMETIC 

•v 


BY 


ALVA  w.  STAMPER,  PH.  D. 

HBAI)  OF  THK   DEPARTMENT  OF   M ATHKM ATICS,   STATE  NORMAL  SCHOOL 
CHK'O,   (,'ALJFoRMA 


Normal  Junior  Press 

Chico,  California 

1909 


CONTENTS 

PAGE 

General  Plan       ... 

Notation  and  Numeration  ...  3 

Subtraction 

•  .            4 

Division  of  Common  Fractions    ...  6 

Division  of  Decimal  Fractions     .  7 

Mensuration.  -  Area  of  the  Rectangle       .  9 

Surveying.-  Heights          .         .  n 

Keeping  Accounts.  -  Personal  Cash  Account    '.        '.  14 

Lending  and  Borrowing  Money.  -  Bonds  16 


379650 


LESSON  PLANS 


Every  teacher,  every  day,  should  plan  her  lessons  systematically, 
even  if  the  work  is  mere  drill,  but  she  should  always  abandon  any 
part  of  a  plan  when  better  methods  suggest  themselves  during  the 
progress  of  a  lesson. 

In  presenting  new  work,  inexperienced  teachers  are  apt  to  over- 
develop. Be  wary  of  long  tedious  presentations  of  matter  that  is  be- 
yond the  appreciation  of  young  minds,  especially  in  the  lower  grades. 
While  it  is  necessary  for  the  teacher  to  explain  and  develop  new  prin- 
ciples, it  is  more  important  to  follow  this  up  with  sharp  quick  drills, 
for  there  is  no  surer  way  of  finding  out  if  principles  are  understood. 

In  developing  a  working  principle  or  explaining  a  problem,  resolve 
the  same  into  a  series  of  steps  and  make  sure  that  the  class  can  do 
the  same. 

The  following  plan,  which  may  be  used  in  presenting  new  work  in 
any  of  the  school  subjects,  lends  itself  readily  to  preparatory  lessons 
in  arithmetic. 


LESSON 

(General   Plan) 

a.  Unit  of  Instruction:   This  may  be  a  topic  with  or  without   sub- 
topics. 

b.  Special  Phase  of  this  Unit:  This  may  be  a  sub-topic  or  a  special 
kind  of  problem. 

c.  Devices  and  Materials. 

1 


Method  of  Procedure 

1.  Aim.  —  As  the  lesson  progresses,    the  class  should  clearly  un- 
derstand the  ends  sought.     The  aims,  if  there    be    more  than  one? 
may    be  distributed  throughout  the    lesson,    not  always   formally. 
Proper  incentives  given  a  class  at  the  right  time  make  toward  good 
results. 

2.  Preparation.—  Prepare  for  the  new  by  reviewing  the  features 
of  the  old  that  are  to  be  used  in  the  development. 

3.  Presentation.—  Introduce  the  new  subject-matter,   relating  it 
to  the  centers  of  association  set  up  in  (2). 

4.  Generalization.      As  a  summation  of  the  conclusions  reached 
in  (3),  deduce  the  working  principle  or  rale. 

5.  Application.  -  This  tests  the  pupil'3  understanding  of  the  prin- 
ciples stated  in  (4). 

The  last  four  of  the  five  formal  steps  in  the  above  plan  can  not  be 
clearly  defined  in  treating  some  topics.  The  arrangement  given  must 
be  considered  as  elastic. 

The  aim  of  the  teacher  is  usually  different  from  the  one  given  the 
class,  being  broader  and  sometimes  extending  through  several  les- 
sons. 

The  following  lesson  plans  are  to  be  considered  as  suggestions. 
No  two  teachers  can  present  any  topic  in  the  same  way.  Neither 
can  one  teacher  easily  duplicate  any  one  of  her  own  lessons  as  giv- 
en. Much  depends  upon  the  treatment  of  the  preceding  material 
that  forms  a  basis  for  the  new  topic.  Teachers  are  naturally  in- 
fluenced in  their  methods  of  procedure  by  the  texts  they  are  using. 

We  precede  each  Lesson  Plan  by  a  brief  summary  of  the  review 
that  bears  on  the  new  lesson.  Some  of  this  preliminary  work  may 
have  been  done  weeks  before  in  relation  to  other  topics  and  some 
may  have  been  recently  developed  as  necessary  steps  leading  to  the 
new  lesson.  Inexperienced  teachers,  especially,  should  thus  analyze 
the  new  work  in  relation  to  the  old.  They  should  also  reduce  the 
number  of  new  things  in  any  one  lesson  to  a  minimum. 


Lesson  I 

Unit  of  Instruction:  Notation  and  Numeration. 

Special  Phase  of  this  Unit:    Numbers  of  one,  two,  and  three  digits. 

Devices  and  Materials. 

Colored  sticks  or  toothpicks.  Rubber  Bands.  A  box  with  three 
compartments  to  correspond  to  the  units',  tens',  and  hundreds'  or- 
ders; numbers  to  be  built  on  the  box  lid.  In  lieu  of  a  box,  a  dia- 
gram may  be  drawn  on  a  table. 

Basis  for  the  Lesson. 

a.  Ability  to  read  numbers  of  one,  two,  and  perhaps  three  digits. 

b.  Ability  to  write  the  same. 

c.  Ability  to  assign  number  values  to  groups  of  objects. 

Method  of  Procedure 

1 .  Pupil's  Aim. 

We  wish  to  learn  more  about  the  reading  and  writing  of  numbers 
so  as  to  understand  how  to  add  long  columns.  We  will  first  build 
some  numbers  out  of  sticks. 

2.  Preparation. 

If  necessary  review  (a),  (b, )  (c)  above. 

3.  Presentation. 

a.  Build  numbers  on  lid  of  numeration   box.    Begin   with   loose 
sticks  in  units'  box,  the  other  boxes  being  empty.    Have   a   pupil 
count  out  17  sticks.     Place  band  around  10  of  them.     Build  the 
number.     Build  26,  34,  etc.,   and  have  them  read. 

Write  these  numbers  on  the  board  as  they  are  built. 
Teach  names  of  units'  and  tens'  orders. 

b.  Write  29,  37,  etc.  on  the  board.    Have   pupils   read   and  build 
them. 

c.  Build,  read,   and  write  124,   236,  etc.,    thereby  teaching  hun- 
dreds' order. 

d.  Pupils    write    on   the  board  numbers  given  by  the  teacher. 
Enumerate  the  orders. 


4.  Generalization. 

As  a  result  of  the  above  development  the  pupils  are  to  learn: 

a.  The  names  of  the  different  orders. 

b.  Any  order  must  contain  no  number  greater  than  9. 

c.  In  a  number  like  326,  the  2  represents  two  bundles  or  groups, 
ten  units  like  those  of  the  6  in  each  bundle  or  group. 

The  3  represents  three  bundles  or  groups,  ten  units  like  those 
of  the  2  in  each  bundle  or  group.  The  3  also  represents  three  hun- 
dred of  the  units  like  those  of  the  6. 

5.  Application. 

This  has  been  partially  given  under  Presentation.  Since  the 
pupils  can  probably  already  read  and  write  nuinbers  of  three  digits 
quite  readily,  there  is  no  need  of  continuing  this  farther.  Ques- 
tion them  on  the  points  emphasized  under  Generalization. 

(  Take  up  "carrying"  in  column  addition  in  thanext  day's  lesson 
as  a  farther  application. ) 

Note:  It  may  be  possible  to  introduce  thousands'  order  in  the 
above  lesson.  In  a  later  lesson  the  pupils  should  learn  about  the  dif- 
ferent periods. 


Lesson  II 

Unit  of  Instruction:  Subtraction. 

Special  Phase  of  the  Unit:      Use  Addition  Method  where  "carry- 
ing" is  involved. 
Basis  for  the  Lesson. 

a.  Ability  to  make  change  by  the  making-up  method,  as  com- 
monly used  in  stores.     Very  simple  examples  requiring  only  men- 
tal work. 

b.  Ability  to  subtract  in  an  example  like  849 

217  ,  where  the   orders 

in  the  subtrahend  are  less  than  the   corresponding  orders   in   the 
minuend. 

c.  The  habit  of  subtracting  in   an  example  like  the  above    by 


saying  7  and  2  are  9. 
d.     Ability  to  add  columns  involving  "carrying." 

Method  of  Procedure 

1.  Pupil's  Aim. 

How  many  can  subtract  2013 

1984     ? 
We  will  now  learn  how  to  subtract  such  numbers. 

2.  Preparation. 

a.  A  few  written  examples  based  on  (c)   and   (d)   above  may 
be  necessary. 

b.  Quick  drill  on  some  of  the  harder  making-up  combinations, 
especially  those  that  will  be  used  bafore  the  lesson  is  over.  Thus: 
9  and  what  are  17?  8  and  what  are  13?  Etc. 

3.  Presentation. 
Let  us  subtract  93 

68. 

Is  this  different  from  the  examples   we  have   been  working? 

How  does  it  differ? 

Does  any  number  added  to  8  give  3? 

What  is  the  first  number  larger  than  8  that  ends  with  a  3? 

8  and  what  make  13? 

Where  do  we  write  the  5? 

How  many  to  carry?  (Why  do  we  say  "carry?") 

How  many  can  tell  to  what  figure  in  the  second  column  we  add 
the  1  carried?  ( Since  the  93  is  the  sum  of  the  68  and  the  desired 
remainder,  we  must  add  the  1  to  the  6.) 

We  then  look  at  the  6  and  think  7. 

What  do  we  next  say? 

What  figure  do  we  write  to  the  left  of  the  5?    The  answer? 

Erase  the  answer  and  repeat  the  subtraction. 

Next  erase  the  93  and  the  line  beneath  the  68  and  add  68  and 
25,  placing  the  sum  above  the  68. 

Erase  the  25  and  repeat  the  subtraction. 

Ex.  Subtract  58  from  83. 


4.  Generalization. 

a.  If,  as  in  the  above  example,   the  8  is  greater  than  the  3,  we 
must  think  of  13  in  the  place  of  the  3.    If  a   2   were   above  the  8, 
think  of  a  12  in  its  place.  Etc. 

b.  Add  the  1  "to  carry"  to    the    next   figure   in  the    subtra- 
hend to  the  left. 

c.  After  each  of  these  changes  are  made   subtract  as  in    prev- 
ious work. 

5.  Application. 

a.  Write  a  list  of  examples  on  the  board  and  see  if  the  class  can 
follow  the  directions  given  in  (a)  and  (b)  above,   not  necessarily 
writing  down  the  answers  at  first. 

Thus  write   85 

27 .  The  pupils  say:  "Think  of  15  in  the  place  of  5. 
Add  1  to  2,  thinking  3  in  the  place  of  2''. 

b.  Now  give  examples  for   subtraction,  requiring   statements 
like  the  above  in  each  step  of  the  process. 

c.  Subtract  numbers  of  three  or  more  digits  each. 


Lesson     III 

Unit  of  Instruction:  Division  of  Common  Fractions. 

Special  Phase  of  the  Unit:     The  rule  for  inverting  the  divisor. 

Basis  for  the  Lesson. 

a.  Multiplication  of  fractions. 

b.  Division  of  fractions  with  unlike  denominators  by  reducing 
to  common  denominators. 

Method  of  Procedure 

1.  Pupil's  Aim. 

We  wish  to  learn  a  shorter  way  of  dividing  fractions  that  have 
unlike  denominators. 

2.  Preparation. 

Give  some  examples  as  in  (a)  and  (b)  above. 
Thus:     Multiply  |  by  i.     Divide  t  by  J. 


7 

3.  Presentation. 

a.  Divide  -J  by  -Jj  just  as  we  have  been  doing. 
Thus  H   |=  A,  -v-  ,*,   -    |i  ==1|. 
Therefore     J +  »=!£. 

b.  Work  this  example  in  multipication:     Multiply  £  by  f.      We 
get  |-  X"|  ==J       U.  Therefore  -}  x  f  =  H. 

c.  Since  f  divided  by  |  gives  li  and  since  f  multiplied  by  f  gives 
the  same  answer,    then  f   divided  by  #  must  equal  f  multiplied  by 
i   which  is   ;i  inverted. 

d.  Divide  £  by  f  by  reducing  to  a  common  denominator.  Multiply 
*  by  J  and  see  if  the  answers  agree. 

4.  Generalization. 

Therefore,  in  order  to  divide  a  fraction  (or  whole  number)  by 
a  fraction,   multiply  the  dividend  by  the  divisor  inverted. 

Thus:  :J  -f  jj  —  ?  X|  =^  .   We  do  not  need  any  longer  to  reduce 
fractions  to  common  denominators  when  dividing. 

5.  Application. 

a.  Work  examples  of  the  above  type. 

b.  And  ethers  where  either  dividend  or  divisor  are  mixed  num- 
bers. 

c.  Also  choose  the  dividend  a  whole  number. 


Lesson  IV 

Unit  of  Instruction:  Division  of  Decimals. 

Special  Phase  of  the  Unit:     The  divisor  a  decimal. 

Basis  for  the  Lesson. 

a.  Division  of  integers,   the  quotient  a  decimal.     Thus:    Divide 
135  by  25. 

This  necessitates  the  use  of  a  point  at  the  right  of  the  divi- 
dend and  the  annexing  of  ciphers. 

Also  the  proper  placing  of  the  first  digit  in  the  overhead 
quotient. 


And  the  placing  of  the  point  in  the  quotient  above  the  point 
at  the  right  of  the  dividend. 

b.  Multiplying  a  number  by  10,   100,  etc.    moves  the  point  one, 
two,  etc.  places  to  the  right. 

c.  Multiplying  both  divisor  and  dividend   by  the  same   number 
does  not  change  the  value  of  the  quotient. 

d.  Multiplying  the  dividend  multiplies  the  quotient,    and    multi- 
plying the  divisor  divides  the  quotient. 

Method  of  Procedure 

1.  Pupil's  Aim. 

Ex.  If  a  boy  earns  $  .65  a  day,  how  many  days  will  it  take  him 
to  earn  $7.80?  In  attempting  to  divide  7.80  by  .65  the  class  sees 
the  need  of  learning  to  divide  by  a  decimal  divisor. 

2.  Preparation. 

Review  such  of  (a),  (b),  (c)  above  as  is  necessary. 

3.  Presentation. 

Ex.  Divide  7.80  by  .65. 

How  does  this  example  differ  from  our  previous  examples  in  the 
division  of  decimals? 

How  can  we  make  the  divisor  a  whole  number,  keeping  the 
same  digits?  (Multiply  by  100.) 

How  would  this  affect  the  value  of  the  quotient? 

If  we  multiply  the  divisor  by  100  so  as  to  get  a  whole  number 
as  a  divisor,  what  else  must  we  do  so  as  not  to  get  too  small  a 
quotient?  (Multiply  the  dividend  by  100.) 

This  leads  to  the  statement  that  we  first  move  the  point  in  the 
divisor  two  places  to  the  right  so  as  to  rid  the  divisor  of  decimals, 
and  then  move  the  point  in  the  dividend  the  same  number  of 
places  to  the  right.  Two  points  indicate  the  new  positions  of  the 
points  as  shown  below.  (A  caret  is  perhaps  better. ) 

1    . 
.65..    7.80.. 

Next  plnce  the  1  of  the  quotient   over   the  proper   digit  in  the 


dividend   (the  8). 

Write  the  next  digit  of  the  quotient  over  the  0  and  place  the 
point  of  the  quotient  above  the  new  point  in  the  dividend. 

4.     Generalization. 
We  proceed  mechanically  as  follows: 

a.  Move  the  point  in  the  divisor  to  the  right  so  as  to  rid  it  of 
decimals. 

b.  Move  the  point  in  the  dividend  the  same  number  of  places  to 
the  right. 

c.  Place  the  first  digit  of  the  quotient  as  in  ordinary  division. 

d.  Place  the  point  in  the  quotient  over  the  new  point  in  the  divi- 
dend.   In  short  division  the  quotient  and  its  point  may  be  written 
beneath  the  dividend. 

5.     Application. 

a.  Work  other  examples  as  above,    where   the   quotient  is  a 
whole  number. 

b.  Examples  where  the  quotient  is  a  decimal. 

c.  Examples  where  ciphers  must  be  annexed  to  the  right  of 
the  dividend  to  accommodate  the  moving  of  the  point. 


Lesson   V 

Unit  of  Instruction:  Mensuration. 

Special  Phase  of  the  Unit:     The  area  of  a  rectangle. 

Devices  and  Materials. 

Drawing  of  rectangle  on  the  board,  divided  as  described  below. 

A  rectangular  board,  say  4"  by  9",  marked  into  square  inches. 
(  This  may  be  used  in  another  lesson  in  finding  the  area  of  a 
parallelogram.) 

Basis  for  the  Lesson. 

a.  Names  of  common  geometric  figures  and  their  chief  charac- 
teristics, especially  points  of  similarity  and  dissimilarity  between 
the  square  and  the  rectangle. 


10 


b.  Ability  to  construct  squares  and  rectangles,  using  preferably 
the  draughtman's  triangles  and  the  scale,  or  graduated  rule. 

Method  of  Procedure 

1.  Pupil's  Aim. 

How  many  of  you  have  fathers  that  own  farms?  If  each  of  your 
fathers  were  to  sell  his  farm,  how  would  he  figure  what  it  is 
worth  to  him?  This  leads  to  the  question  of  area  in  general  and 
that  of  a  rectangle  in  particular. 

2.  Preparation. 

Perhaps  no  special  review  is  necessary,  but  (a)  and  (b)  above 
should  be  understood. 

3.  Presentation. 

Draw  on  the  board  a  rectangle  8".  by  14" ,  the  longer  sid  2  or, 
base,  being  horizontal. 

Mark  the  sides  into  divisions  of  1"  each. 

Draw  a  horizontal  line  1"  above  the  base  so  as  to  form  a  strip 
1"  wide  and  14"  long. 

Divide  this  strip  into  square  inches. 

How  many  square  inches  in  this  strip? 

How  many  strips  like  this  could  we  draw?  Draw  them,  but  no 
more  squares. 

How  many  square  inches  in  8  strips?  Ans.  8  X  14  sq.  in.,  or  132 
sq. in. 

Into  how  many  square  inches,  then,  may  this  rectangle  be 
divided? 

4.  Generalization. 

Let  us  erase  all  the  lines  drawn  in  our  rectangle.  Who  can  tell, 
without  trying  to  picture  the  squares,  how  we  get  132  as  the  num- 
ber of  square  inches  that  make  up  the  rectangle?  [We  multiply 
14  by  8,  or  8  by  14.] 

How  do  we  find  the  number  of  square  inches  in  a  rectangle  16" 
by  20"? 

Introduce  the  term  '  'area. ' ' 

Find  the  area  of  a  rectangle,  the  lengths  of  whose  sides  are 
expressed  in  feet. 


11 

I 

The  following  should  be  emphasized: 

a.  The  length  and  breadth  must  be  expressed  in  like  units. 

b.  The  unit  of  area  depends  upon  the  unit  of  length.    (  How?  ) 

c.  Use  of  term  "dimensions.'"' 

d.  The  area  of  a  rectangle  is  found  by  multiplying  the  number 
of  units  of  length  by  the  number  of  units  of  breadth.    Or  it  is  the 
product  of  the  two  dimensions  of  the  rectangle. 

e.  The  term  "area"  defines  the  number  of  square  units  in  the 
figure. 

[Do  not  say  that  the  area  is  5"  times  4",  for  inches  times  inches 
cannot  give  square  inches.  The  multiplier  must  always  be  ab- 
stract. Say  5  times  4  gives  20,  the  number  of  square  inches.] 

5.     Application. 

a.  Give  other  examples  involving  different  linear  units,    requir- 
ing only  mechanical  work. 

b.  Have  a  rectangle  drawn  and  also  its  unit  of  area. 

c.  Find  the  area  of  the  school-room  floor,  measuring  its  dimen- 
sions to  the  nearest  foot  and  half  foot. 

Note:  In  the  following  lessons  have  plans  of  irregular  fig- 
ures drawn  to  scale,  choosing  figures  that  can  be  divided  into 
rectangles.  Compute  perimeter  and  area.  The  ground  plan  of 
the  school  building  may  give  suitable  data. 


Lesson  VI 

Unit  of  Instruction:  Surveying. 

Special  Phase  of  the  Unit:      Heights. 

Devices  and  Materials. 
A  measuring  tape.  A  short  stick  or  lead  pencil. 

Basis  for  the  Lesson. 

a.  An  understanding  of  the  terms:  Vertical  line,  perpendicular, 
horizontal  line,  right-angled  triangle,  parallel  lines. 

b.  Ability  to  make  drawings  of  the  same. 

c.  Parallel  lines  are  everywhere  equidistant. 


12 


d.  The  proportions  between  corresponding  sides  in  similar  tri- 
angles. This  is  not  necessary  in  case  the  teacher  wishes  to 
give  these  as  rules  in  connection  with  the  lesson,  but  it  is  neces- 
sary that  the  pupils  be  able  to  solve  for  x  in  proportions  like 
3  :8  --=  6  :x. 

Method  of  Procedure 

1.  Pupil's  Aim. 

How  many  have  heard  of  lumbermen  finding  the  height  of  trees 
when  estimating  timber?  We  shall  learn  to-day  how  to  find  cer- 
tain heights  without  measuring  them. 

2.  Preparation. 

Review  the  terms  given  under  (a)  above  and  such  of  (d)  as  have 
been  taught. 

3.  Presentation. 

We  wish  to  find  the  height  of  this  room.  (Specific  aim.) 
[As  the  measurements  are  taken  let  a  drawing  be  made  on  the 
board,  as  nearly  to  scale  as  possible.] 

We  shall  find  the  distance  from  a  corner  of  the  ceiling  to  the 
corresponding  corner  of  the  floor  [the  line  joining  the  points  rep- 
resented by  E  and  G]  . 


•  E 


H 


13 

A  pupil  stands  in  the  position  AB,  his  eye  at  point  A,  and  his 
arm  extended  so  that  when  he  sights  over  the  top  of  the  stick 
CD  held  vertically  in  his  hand,  he  sees  the  corner  of  the  ceiling 
[at  E] .  At  the  same  time  he  sights  under  the  lower  end  of  the 
stick  so  that  the  line  of  sight,  AH,  is  horizontal. 

The  stick  CD  has  already  been  measured,  and  is  1  foot  long. 
Measure  AB  (4'  6"),  the  height  of  the  pupil  from  his  eye  to  the 
floor.  Measure  AD  (2  ),  the  distance  from  the  eye  to  the  bottom 
of  the  stick.  Measure  BG  (28'),  the  distance  from  the  feet  of  the 
pupil  to  the  corner  of  the  floor. 

Why  does  AH  equal  BC  ? 

Reach  the  conclusion  that  since  CD  is  |  of  AD,  EH  is  \  of  AH. 
Therefore  EH  14'. 

How  long  is  EG?          EG  =  EH  +  HG  =  14  -f  4'6"   -  18'  6". 

Get  the  value  of  EH  also  from  the  proportion,  AD  :  CD  =  AH  : 
EH.  Substituting  the  proper  lengths,  we  get  2  :  1-^28  :  EH. 
Hence  2  EH  28  and  EH  14. 

4.  Generalization. 

a.  We  measure  first  the  small  stick   that  is  held  in  the  hand. 
Hold  the  stick  in  the  position  described  above.    Measure  the  dis- 
tance from  the  eye  to  the  bottom  of  the  stick.    Measure   the  dis- 
tance from  the  point  on  the  floor   where  the  pupil   stands  to  the 
point  on  the  floor  directly  beneath  the  point  whose  height  is  to  be 
found.  Measure  the  height  of  the   pupil  from  his  eye  to  the  floor. 

b.  See  that  the  pupils  can  tell  how  to  write  the  proportion  as 
given  under  Presentation,    and  that  they  can  finally  tell  how  to 
get  the  required  height. 

5.  Application. 

a.  Substitute  values  in  another  drawing   on  the   board  to   test 
(b)  above. 

b.  Find  another   height  to  test  (a)  and  (b)    in    Generalization 
either  in  this  lesson  or  the  next. 

c.  Have  the  class  draw  to  scale   the  figure  represented  under 
Presentation  for  home  work. 

Note:  CD  and  AD  in  the  above  figure   may  be   in   inches   and 


14 

BG  [or  AH]  in  feet. 
It  is  not  necessary  that  the  bottom  of  the  stick  or  pencil  be  held 

on  a  level  with  the  eye,  but  if  done  a  little  complexity  is  avoided. 
The  use  of  the  45°  right  triangle  provides  an  easy  method  of  find- 
ing heights. 


Lesson  VII 

Unit  of  Instruction:  Keeping  Accounts. 

Special  Phase  of  the  Unit:        First  lesson   in    keeping  a  personal 

cash  account. 
Materials. 

A  small  cash  book,    about  3J"  by  6".      The  right   side   of  each 
page  should  have  lines  for  double  columns  of  figures. 

A  ruler.  Black  and  red  ink. 

Basis  for  the  Lesson. 

a.  Accuracy  in  column  addition. 

b.  Ability  to  use  the  addition  method  of  subtraction  in  writing 
down  a  missing  addend  in  a  column  of  figures  whose  sum    is   set 
down. 

c.  Multiplication  of  decimals  as  related  to  United  States  Money. 

d.  Possibly  multiplication  of  easy  common  fractions. 

Method  of  Procedure 

1.     Preparation. 

a.  On  the  day  previous  each  pupil  is  asked  to  bring  in  a  short 
list  of  amounts  of  money  (real  and  imaginary)  received  by  him  on 
on  specified  dates,  within  the  last  week  or  month. 

b.  Also  of  amounts  (real  and  imaginary)    paid  out  by  him  on 
specified  dates  within  the  same  period. 

c.  The  aim  is  that  the  pupils  learn  how  to  keep  their  own  per- 
sonal accounts. 


15 

2.  Presentation. 

Let  the  children  contribute  as  much  as  possible  to  the  develop- 
ment of  the  lesson. 

a.  What  title  is  to  be  written  in  the  front  of  the  book?  (  ~ — 's 
personal  cash  account.) 

b.  On  which  page  of  the  open  book  do  ,we  write  the  list  of  ex- 
penditures?    (The  credit,  the  right-hand,  side.) 

Why  do  we  credit  cash  that  is  paid  out?  Ans.  Because  the 
person  paying  the  money  must  give  himself  credit  for  doing  this, 
in  his  own  cash  account. 

c.  Why  do  we  debit  cash  that  is  received?    Ans.    Because  the 
person  receiving  the  money  is  indebted  that  much  to  some  one  for 
rendering  him   that  service. 

d.  Show  where  to  write  the  year,    the  terms  Dr.  and  Cr.,  and 
the  items  brought  to  class  as  assigned  the  previous  day. 

e.  Write  original  entries  in  the  first  of  the  double  columns  on 
each  page. 

f.  Write  sub-totals,    balances,    totals,    balances  brought  down, 
amounts  carried   forward,    and  amounts  brought  forward  in  the 
right-hand  columns. 

g.  Plan  to  have  balance  made  in  the  middle  of  the  first  page. 
Balance  the  account  by  the  method  suggested  in  [b]  of  '  'Basis  for 
the  Lesson''  above. 

h.  Write  other  entries  and  write  at  the  bottom  of  each  page  the 
amounts  to  be  carried  forward. 

i.  On  top  of  the  second  open  page  write  the  amounts  brought 
forward.  Enter  other  items. 

j.  Rule  all  lines  in  red  ink.  Write  balance  in  red  ink,  but  not 
the  balance  brought  down  on  the  Dr.  side. 

3.  Generalization. 

This  is  largely  provided  for  in  the  above.  But  emphasize  especi- 
ally: 

a.  The  significance  of  debiting  and  crediting  entries. 

b.  The  object  in  balancing  an  accuant.    The  significance  to  be 
attached  to  the  amount  written  on  ti  c    Cr.  side  that  balances  the 
account. 


16 

4.     Application. 
Continue  the  work  already  begun. 

Note:  The  double  column  page  is  not  necessary,  but  its  use  is 
advisable. 

In  a  later  lesson  teach  the  writing  of  receipts  and  receipted 
bills  in  relation  to  some  of  the  entries. 

Pupils  may  do  a  buying  and  selling  business  among  them- 
selves, using  imitation  money,  and  small  printed  cards  to  repre- 
sent various  kinds  of  merchandise.  Such  money  and  cards  may  be 
purchased  from  the  publishing  houses. 

If  any  business  is  done  on  credit,  open  accounts  with  the  proper 
persons,  thus  introducing  the  use  of  a  ledger. 


Lesson  VIII 

Unit  of  Instruction:  Lending  and  Borrowing  Money. 

Special  Phase  of  the  Unit:      Bonds. 

Materials  and  Devices. 

School,  municipal,  corporation,  or  United  States  bonds,  if  avail- 
able.   Otherwise  a  blank  form  or  copy  of  a  bond. 

Basis  for  the  Lesson. 

Simple  Interest.     Promissory  Notes.     Ordinary  applications  in 
business  arithmetic. 

Method  of  Procedure 

1.  Pupil's  Aim. 

The  teacher  naturally  tells  the  class  that  they  are  to  learn  some- 
thing about  bonds.   A  definite  aim  is  developed  in  the  Presention. 

2.  Preparation. 

A  preliminary  review  is  perhaps  unnecessary. 

3.  Presentation. 

The  city  of   Chico  decided  in  1902  to  install  a   sewer  system   to 


17 

cost  $45,000.    No  funds  were  available. 

In  what  two  ways  could  the  money  have  been  raised?    Ans.  By 
special  tax  and  by  borrowing. 

It  was  decided  to  borrow  the  money. 

When  an  individual  borrows  money,    what  must  he  give  the 
lender?  (  A  promissory  note  .) 

Cities  that  borrow  money  also  give  that  which  is  equivalent  to 
notes,  namely  bonds. 

The  teacher  will  here  bring  out  the  points  of  similarity  and  dis- 
similarity between  notes  and  bonds. 

After  the  borrower  determines  the  rate  of  interest  he  is  willing 
to  pay  and  at  what  intervals  he  wishes  to  pay  back  portions  of 
the  sum  borrowed,  he  usually  advertises  the  bonds  "f or  sale. " 
The  highest  bidder  is  the  person  who  gives  the  borrower  the 
greatest  "bonus"  for  the  privilege  of  lending  him  the  money. 
This  bonus,  or  gift,  to  the  borrower  is  called  the  premium.  The 
borrower  issues  as  many  bonds  as  there  are  times  that  he  pays 
back  portions  of  the  principal.  A  bond  is  said  to  be  redeemed 
when  it  is  returned  to  the  borrower  on  his  paying  the  principal 
named  in  the  bond  together  with  the  interest  due. 

4.  Generalization. 

a.  The  pupils  learn  that  the  bond  has  the  function  of  a  promis- 
sory note. 

b.  They  learn  how  cities  (and  other  corporations)  borrow  money 
and  how  the  money  is  paid  back. 

c.  They  learn  about  paying  premiums  on  bonds  by  the  lender 
and  why  he  does  so. 

5.  Application. 

Ex.  The  City  of  Chico,  California,  on  January  1,  1902,  issued 
bonds  for  $45,000  to  provide  funds  for  installing  a  sewer  system. 
The  bonds  were  forty  in  number  and  of  equal  face  value.  One 
bond  was  to  be  redeemed  on  January  1  of  each  year,  beginning 
with  January  1,  1903.  Interest  was  to  be  at  the  rate  of  5'/  per 
annum,  payable  semi-annually  on  January  1  and  July  1  of  each 
year.  The  bonds  were  sold  at  a  premium  of  $1,100.  Find  the 
total  amount  that  will  have  been  paid  by  the  city  when  all  the 
'bonds  have  been  redeemed,  and  find  the  net  cost  to  the  city. 


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